Inference for the Slope of a Regression Line

A sample scatterplot almost never has a slope of exactly zero, even when x and y are truly unrelated. Random scatter alone can tilt a line one way or another. So before trusting that x predicts y, you test the null hypothesis that the true slope, written beta, equals zero. If you can reject that, you have evidence x actually matters for predicting y, not just in this one sample but in the wider population or process the sample came from.

Regression output gives you everything needed: the slope b, its standard error SE, and from those a t statistic and p-value. The t statistic is t = b divided by SE, measuring how many standard errors the sample slope sits from zero. Degrees of freedom are n - 2, since estimating a line costs two parameters, slope and intercept. The p-value comes from the t-distribution with that df, and small p-values (conventionally below 0.05) reject the claim that beta is zero. A confidence interval for beta takes the same ingredients: b plus or minus a critical t value times SE, giving a range of plausible true slopes.

This mirrors the two-sample t-test from comparing means: both start with a statistic, divide by its standard error to get t, and use a t-distribution to judge how surprising the result is. There you tested whether two population means differed; here you test whether a line's tilt differs from flat. Both also need conditions: here, linearity, independence, roughly normal residuals, and constant spread around the line, checked through a residual plot before you trust the inference at all.